Moore, C.: Predicting non-linear cellular automata quickly by decomposing them into linear ones. Physica D 111 (1998) 27--41  Home/Search   Document Details and Download   Summary   Related Articles   Check

....practice shows that it is rarely the case) that behaviours synthesised in the component semigroups of the wreath product decomposition can be composed to form a valid behaviour in the original semigroup. We require a product whose components can be guaranteed to compose to give global behaviours. Moore (1998) provides such a notion of product and decomposition; however, we need a more generally applicable notion of composition than the one he provides, which is restricted to cellular automata whose rule is a solvable group. Solvability is too strong a property, e.g. in the case of polynomials this ....

Moore, C., (1998). Predicting non-linear cellular automata quickly by decomposing them into linear ones, Physica D, 111, pp. 27-41.

....nilpotency, formal languages, trees and subtrees. Subject Classi cation: 20N05, 68Q70 1 Introduction A body of recent work focuses on the computational complexity of various problems involving algebraic structures, such as evaluating circuits and expressions [2 4] predicting cellular automata [15, 16], solving equations [11] and communication complexity [19] While these algebraic problems are interesting in their own right, they also o er elegant characterizations of some low lying complexity classes, and may even help us prove new separations between them. Most of this work has dealt with ....

....of computational complexity, especially low level parallel complexity classes. For instance, expressions and circuits over solvable groups can be evaluated in the classes ACC 0 and ACC 1 , while over non solvable groups these problems are NC 1 complete and P complete respectively (see [3, 4, 15] for de nitions of these classes and proofs of these results) Similarly, equations over nilpotent groups can be solved in polynomial time, while for non solvable groups this problem is NP complete [11] and for solvable groups quasipolynomial time is believed to suce. Finally, languages de ned ....

C. Moore, \Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones." Physica D 111 (1998) 27-41.

....Turing machines [4] this problem is P complete in general. However, if the cellular automaton has certain algebraic properties, we can predict it much more quickly, in O(log t) or O(log 2 t) time. Thus in special cases the CA prediction problem can be in NC, even for some nonlinear rules [5, 6]. On the other hand, some systems such as lattice gases [7] sandpiles [8] and zero temperature Ising dynamics [9] are P complete to predict, meaning that unless P = NC (in which case all polynomialtime problems are eciently parallelizable) there is no way around simulating them step by step. ....

C. Moore, \Predicting non-linear cellular automata quickly by decomposing them into linear ones." Physica D 111 (1998) 27-41.

....evaluation problem. Then depending on the algebraic properties of (A; such as associativity, commutativity, solvability and so on, Expression Evaluation and Circuit Value can have varying complexities. Previous results for the associative case (groups and semigroups) include the following [3, 7, 22]: Expression Evaluation Circuit Value non solvable NC 1 complete P complete solvable ACC 0 ACC 1 DET In this paper, we will extend these results to non associative groupoids such as quasigroups and loops, and to some extent to groupoids in general. We will show that the idea of ....

....automaton for a polynomial number of steps corresponds to a special case of Circuit Value where the circuit has a periodic structure. Thus these results will also help us tell when there are fast algorithms for predicting cellular automata whose rules correspond to certain groupoids, as in [22, 23]. The paper is structured as follows. In Section 2 we give an introduction to the algebraic terms and concepts we will use. In Section 3 we de ne Booleancompleteness, the ability to express arbitrary Boolean functions as circuits or expressions. We review existing results on solvability in groups ....

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C. Moore, \Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones." Physica D 111 (1998) 27-41.

.... x y 1 0 1 1 w z More generally, say that is ane with respect to an abelian group if it is of the form a b = f(a) g(b) h where f and g are homomorphisms on . The behavior of such rules is easily predictable [7] even if the f s, g s and h s vary in space time [8]. Theorem 18. Two ane CAs, a b = f(a) g(b) h and a b = j(a) k(b) l, commute if and only if the following relations hold: jf = f j; jg kf) fk gj) and kg = gk (4) j k) h) l = f g) l) h (5) Proof. Equation (1) becomes jf(a) jg kf) b) kg(c) j k) h) l = ....

C. Moore, \Predicting nonlinear cellular automata quickly by decomposing them into linear ones." Physica D 111 (1998) 27-41.

....out that predicting cellular automata is P complete in general, since cellular automata exist (e.g. 18] which can simulate universal Turing machines. On the other hand, NC algorithms exist for Eden growth [20] the Lorentz lattice gas [31] and cellular automata with certain algebraic properties [29, 30]. Even if a speedup to polylogarithmic time isn t possible, we might still hope for a polynomial speedup predicting physical time t in O(t ) parallel time for some 1. For instance, in Ref. 32] it was shown that though ordinary DLA is P complete, on average it can be parallelized to ....

C. Moore, \Predicting non-linear cellular automata quickly by decomposing them into linear ones." Physica D 111 (1998) 27-41.

....cantly easier. Languages recognized by solvable groups have simple combinatorial descriptions [18, 21] and circuits over them can be evaluated quickly in parallel [2, 3] Similarly, cellular automata de ned with polyabelian operations can be predicted much more quickly than by explicit simulation [14]. Thus the algebraic properties of a groupoid are intimately linked to its computational complexity. This paper is organized as follows. Section 2 gives an introduction to the algebraic terms and concepts we will use. In Section 3 we de ne the functional closure of a groupoid, and show that simple ....

....1 t 1 t 2 Then G=N has an Abelian normal subgroup, and so on; by induction G is polyabelian. If t 1 t 2 = t 1 t 2 so that h = 0, then T is a subgroup of G isomorphic to G=N , the quasidirect product reduces to the semidirect product on groups, and G is a split extension of N by T [20] In [14] we de ned polyabelianness with semidirect products only, in which case any solvable group is a subgroup of a polyabelian group by iterating wreath products. Lemma 5.7 Nilpotent loops are polyabelian. Proof. Let G be a nilpotent loop with center Z(G) Then the local operation in G=Z(G) Z(G) ....

C. Moore, \Predicting non-linear cellular automata quickly by decomposing them into linear ones." Physica D 111 (1998) 27-41.

....together k = 2r sites, we can transform any CA into one with r 0 = 1=2. Here r = 2 and k = 4. This can be a fruitful point of view from which to study CAs. Depending on ffl s algebraic properties, we can make statements about how much parallel or serial computation is needed to predict the CA [4, 5], its reversibility or surjectivity [1, 3] or its periodic behavior [7] In fact, any CA is equivalent to one with r = 1=2 through the following block transformation. Treat blocks of k sites as single sites of another CA rule, with a larger alphabet A k and a smaller radius r 0 = r=k (if k ....

....binary operation, the block algebra of the original CA. We would like to know, then, to what extent block transformations can simplify the analysis of CAs. Can non linear CA rules on several site neighborhoods be equivalent to binary algebras with nice algebraic properties, such as those shown in [4, 5] to allow efficient prediction of the CA We will show that, under many circumstances, they cannot. More precisely, if a CA s block algebra is in one of four large classes of algebras (which include groups, monoids, common types of non associative algebras, and commutative algebras in general) ....

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C. Moore, "Predicting non-linear cellular automata quickly by decomposing them into linear ones." To appear in Physica D, Proceedings of the International Workshop on Lattice Dynamics..

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Moore, C.: Predicting non-linear cellular automata quickly by decomposing them into linear ones. Physica D 111 (1998) 27--41

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